Problem: What is the coefficient of $x^3$ when $$x^4-3x^3 + 5x^2-6x + 1$$is multiplied by $$2x^3 - 3x^2 + 4x + 7$$and the like terms are combined?
Explanation: Instead of expanding the entire product, we can look only at terms that will multiply to give $x^3$.  We know that: $$x^3=x^3\cdot 1=x^2\cdot x=x\cdot x^2=1\cdot x^3$$Knowing this, the $x^3$ term in the expansion will be the sum of these four terms: $$(-3x^3)(7)+(5x^2)(4x)+(-6x)(-3x^2)+(1)(2x^3)$$We simplify to find: \begin{align*}
&(-3x^3)(7)+(5x^2)(4x)+(-6x)(-3x^2)+(1)(2x^3)\\
&\qquad=-21x^3+20x^3+18x^3+2x^3\\
&\qquad=\boxed{19}x^3.
\end{align*}